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 May 20th, 2017, 06:49 PM #1 Senior Member   Joined: Nov 2015 From: hyderabad Posts: 232 Thanks: 2 Problems on Series 1. The largest Interval $I$ such that the series $\displaystyle \sum_{n=1}^{\infty} \frac{x^n}{\sqrt{n}}$ converges whenever $\displaystyle x \in I$ is equal to A)$[-1,1]$ B)$[-1,1)$ C)$(-1,1]$ D)$(-1,1)$ My answer : Options A or B 2. Let $\displaystyle \sum a_n$ be a convergent series. Let $b_n=a_{n+1} - a_n$ for all $\displaystyle n \in N$. Then A) $\displaystyle \sum b_n$ should also be convergent and $\displaystyle (b_n) \rightarrow 0$ as $\displaystyle n \rightarrow \infty.$ B) $\displaystyle \sum b_n$ need not be convergent but $\displaystyle (b_n) \rightarrow 0$ as $\displaystyle n \rightarrow \infty.$ C) $\displaystyle \sum b_n$ is convergent but $(b_n)$ need not tend to zero as $\displaystyle n \rightarrow \infty.$ D) None of the above statements is true. My answer : Option A 3. Which of the following series converge ? A) $\displaystyle \sum_{n=1}^{\infty} (\frac{\log n}{n^{1+2\epsilon}}$ B) $\displaystyle \sum_{n=1}^{\infty} (\frac{(\log n)^2}{n^{1+2\epsilon}}$ C) $\displaystyle \sum_{n=1}^{\infty} (\frac{n^2+1}{n^3+n})$ D) $\displaystyle \sum_{n=1}^{\infty} (1+\frac{1}{n})^n$ I checked Option C, it is convergent and Option D is not convergent. Kindly check if Option A or B are Convergent ? Help me to solve series with logarithms. Kindly check and let me know If I'm wrong. Thanks Last edited by skipjack; May 20th, 2017 at 09:43 PM.
 May 20th, 2017, 10:25 PM #2 Global Moderator   Joined: Dec 2006 Posts: 20,379 Thanks: 2011 1. (B) 2. (A) 3. The parentheses don't balance in (A) and (B), and ϵ is undefined. (C) and (D) are divergent.
May 20th, 2017, 10:41 PM   #3
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 Originally Posted by skipjack 1. (B) 2. (A) 3. The parentheses don't balance in (A) and (B), and ϵ is undefined. (C) and (D) are divergent.
Sorry to confuse you. Can you please check the below series now ?

A) $\displaystyle \sum_{n=1}^{\infty} (\frac{\log n}{n^{1+2e}})$
B) $\displaystyle \sum_{n=1}^{\infty} (\frac{(\log n)^2}{n^{1+2e}})$

May 20th, 2017, 10:56 PM   #4
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Quote:
 Originally Posted by skipjack 1. (B) 2. (A) 3. The parentheses don't balance in (A) and (B), and ϵ is undefined. (C) and (D) are divergent.
Can you suggest some videos or books on series where I can get details of how to check if a series is convergent or divergent and the range of n as well as for the values or a.

It's confusing for me to estimate a series converges or not.

Thanks

 May 21st, 2017, 08:44 AM #5 Math Team   Joined: Dec 2013 From: Colombia Posts: 7,622 Thanks: 2611 Math Focus: Mainly analysis and algebra C is precisely $\sum \frac1n$ and thus divergent. The terms of D tend to $e$ and not $0$, so that is divergent.

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